Principles of Fluorescence Spectroscopy

144 TIME-DOMAIN LIFETIME MEASUREMENTS
4.11.3. Stretched Exponentials
A function similar to the lifetime distributions is the
stretched exponential
(4.40)
In this expression β is related to the distribution of decay
times. The function is not used frequently in biophysics but
is often found in studies of polymers when one expects a
distribution of relaxation times. In a least-squares fit, β and
τ would be the variable parameters.
4.11.4. Transient Effects
In many samples the intensity decay can be non-exponential
due to phenomena which occur immediately following
excitation. This occurs in collisional quenching and in resonance
energy transfer. In the presence of a quencher, a fluorophore
that displays an unquenched single-exponential
lifetime will decay according to
(4.41)
In this expression b depends on the quencher concentration
and diffusion coefficient. One can fit such decays to the
multi-exponential model, but one would then erroneously
conclude that there are two fluorophore populations. In this
case a single fluorophore population gives a non-exponential
decay due to rapid quenching of closely spaced fluorophore–quencher
pairs.
Resonance energy transfer (RET) can also result in
decays that have various powers of time in the exponent.
Depending on whether RET occurs in one, two, or three
dimensions, t can appear with powers of 1/6, 1/3, or 2
respectively. Hence we see that intensity decays can take a
number of forms depending on the underlying molecular
phenomenon. In our opinion it is essential to analyze each
decay with the model that correctly describes the samples.
Use of an incorrect model, such as the multi-exponential
model, to describe transient effects, results in apparent
parameter values (α i and τ i ) that cannot be easily related to
the quantities of interest (quencher concentration and diffusion
coefficient).
4.12. GLOBAL ANALYSIS
I(t) I 0 exp(t/τ) β
I(t) I 0 exp(t/τ 2bt 1/2 )
In Section 4.10 we indicated the difficulties of resolving the
decay times and amplitudes in a multi-exponential decay.
The parameters in the various decay functions are correlated
and difficult to resolve. The resolution of correlated
parameters can be improved by the use of global analysis.
200–205 The procedure is to combine two or more experiments
in which some of the parameters are the same in all
measurements, and some are different. This can be illustrated
by the emission spectra in Figure 4.53. A non-global
experiment would be to recover the values of α i and τ i from
the intensity decay collected at 380 nm, where all three fluorophores
emit. A global experiment would be to measure
the intensity decays at several wavelengths, say 360, 380,
400, and 420 nm. The multiple intensity decay curves are
then analyzed simultaneously to recover the τ i values and
the α i (λ) values. The τ i values are assumed to be independent
of emission wavelength. In the case of global analysis
the calculation of χ R 2 extends over several data sets. The
global value of χ R 2 is given by
χ 2 R 1
ν ∑ λ
∑ n
k1
[Iλ c(t k) I λ (t k) 2
I λ (t k)
(4.42)
where the additional sum extends over the files measured at
each wavelength (λ). For the fitted functions the α i values
are different at each wavelength α i (λ) because of the different
relative contributions of the three fluorophores. The values
of τ i are assumed to be independent of emission wavelength
since each fluorophore is assumed to display a single
exponential decay.
It is easy to see how global analysis can improve resolution.
Suppose one of the intensity decays was measured at
320 nm. This decay would be almost completely due to
indole (Figure 4.53), and thus would determine its lifetime
without contribution from the other fluorophores. Since
there is only one decay time, there would be no parameter
correlation, and τ 1 would be determined with good certainty.
The data at 320 nm will constrain the lifetime of indole
in data measured at longer wavelengths and in effect
decrease the number of variable parameters at this wavelength.
Even if the choice of wavelengths only partially
selects for a given fluorophore, the data serve to determine
its decay time and reduce the uncertainty in the remaining
parameters.
Global analysis was used to recover the lifetimes
across the emission spectrum of the three-component mixture,
using the decays measured from 360 to 440 nm. The
lifetime χ R 2 surface for the three decay times is shown in
Figure 4.56. The expected decay time was recovered for
each of the components. However, even with a multi-wave